Fluid-structure interaction analyses of compressors and other reciprocating machinery often involve contact and complex, dynamic boundary conditions. Accurate analyses of such problems require robust algorithms that can account for the coupling between nonlinear structures and Navier-Stokes fluid flow. ADINA FSI offers the reliability and flexibility to accurately model complex FSI problems.

In this Brief, we present an illustrative example of a piston with suction reed valve which incorporates fully coupled FSI, contact, and gap conditions. ADINA’s gluemesh and leader-follower features are also used. We consider some of the interesting results and offer an explanation and further insight using analytical expressions and ADINA’s potential-based element. Finally, we give a step-by-step tutorial for the FSI analysis.

Consider the simplified piston with reed valve geometry shown in Figure 1 below. The topmost surface represents the reciprocating piston, with its motion indicated by dashed arrows. In this configuration, fluid is free to flow between the inlet and the piston chamber. When closed, the valve lies flat on the valve seat, thereby blocking flow. The time-varying upward motion of the piston is prescribed.

Figure 1 Schematic of piston with reed valve

Figure 2 shows some important features of the FSI model. Notably, a thin fluid gap (exaggerated in the figure below) is placed immediately beneath the valve. The fluid is modeled with a low speed compressible, turbulent K-ε formulation. The fluid domain is represented using 4 subdomains corresponding to the inlet and three additional layers as shown below.

Figure 2 Exploded view and side view of fluid domain. The thickness oflayer 1 and layer 2 has been exaggerated

Solid mesh

The reed valve is meshed using two separate element groups. The region in green is meshed using high order 27-node solid elements and the region in red uses less expensive 8-node solid elements. Glue mesh is used to connect the meshes, as denoted by the blue line.

Figure 3 Solid mesh with 2 element groups connected by glue mesh

Contact conditions

The problem involves contact between the valve seat and the underside of the reed valve. The valve seat is the target contact surface and it is assumed to be rigid.

Figure 4 Contact surfaces used in the structural model. Note the hole in the contact surface on the valve seat corresponding to the inlet channel

We use a contact compliance of 1.0 x 10-4 to prevent spotty contact. A contact offset is applied to prevent the thin fluid layer (layer 1) between the valve and the valve seat from compressing to zero volume when the valve is closed.

Figure 5 Modeling the space between the reed valve and valve seat

Flow conditions

This model uses a moving wall boundary condition for the piston, FSI boundaries on all valve surfaces, and gap conditions, see Figure 6. Note the location of the gap condition within the thin fluid region just beneath the valve. The ADINA gap condition is used to connect or disconnect two fluid domains. Fluid can flow across the gap when the gap area exceeds a threshold (i.e., as the valve begins to open), and flow is stopped when the gap area falls below another threshold (i.e., when the valve is closed). As the valve opens and closes, the gap condition will open and close to permit or block fluid flow, as appropriate.

Initial conditions are defined at the inlet and within the piston chamber. The initial pressures at the inlet and within the chamber are 0.113 MPa and 1.47 MPa, respectively.

Figure 6 Locations of gap, FSI and moving wall boundary conditions

ALE moving mesh with the leader-follower option

The leader-follower feature is a useful option to help control the quality of moving meshes. This feature constrains the movement of the moving mesh boundary lines using the two end points of the boundary line. In this model, two "rings" of leader-follower pairs are defined to control fluid mesh distortion as the valve opens and closes as shown in Figure 7.

Figure 7 Location of leader-follower pairs

Figure 8 shows the ALE moving mesh as the valve opens and closes.

Figure 8 ALE moving mesh as valve opens and closes

Results

ADINA FSI predicts that the valve will open three times as the piston displaces, and that the pressure near the top of the piston chamber fluctuates about 0.113 MPa; see Figures 9 and 10. Note that the piston has nearly reached its maximum height the second time that the valve opens, so the second maximum tip displacement is lower.

Figure 9 Time-varying vertical displacements at the reed valve tip

Figure 10 Pressure near the top of the cylinder; detail of fluctuations around 0.113 MPa shown in inset

Validation

With increasing problem complexity, model validation becomes more important; we seek to assure that the predicted results make physical sense. For example, one might ask the following questions:

Why does the valve open and close multiple times?

Why does the piston chamber pressure sometimes exceed the inlet pressure after the valve opens?

The above effects are observed because the period of the piston motion is of the same order of magnitude as the natural period of the system. If the piston motion was slow as compared to the natural frequencies, the valve would not open and close multiple times and the chamber pressure would not fluctuate.

Hand calculations can be performed to estimate the natural frequencies of the system and hence validate the FSI results. The fundamental period of vibration of a cantilevered beam (ignoring added mass effects) is given by

where E is the Young’s modulus, ρ is the density, L is the length, and h is the thickness of the cantilevered beam. Using the above approximation, we obtain a period of ~5 ms. The period of the valve motion predicted by ADINA FSI is ~6 ms, see Figure 9.

To understand why the pressure fluctuates in the piston chamber, we consider the system as a Helmholtz resonator. The fundamental period of vibration for a Helmholtz resonator is

where a is the wave speed of a pressure wave through the fluid, V is the volume of the piston chamber, and c is the inlet conductance given by

where A is the inlet area and L is the inlet length. Using the above approximation, and assuming the valve has just opened, we obtain a period of ~1 ms. The period of piston chamber pressure fluctuations predicted by ADINA FSI is ~2 ms, see Figure 10. We note that the fluctuation speed due to resonance is much smaller than the speed of a pressure wave through the fluid.

These hand calculations validate the ADINA FSI results and give physical insight into the problem. Notably, we observe that inlet length affects the inlet conductance and hence the piston chamber pressure fluctuations. For this reason, it is important that the inlet length is not shortened in the model.

Finally, a better approximation of the system’s natural frequencies can be obtained using ADINA’s potential-based element. In this analysis, the natural frequencies of the combined fluid-structure system are determined. The first period of vibration of the combined system is 4.9 ms (very close to 1st structural period without fluid), and the second period of vibration is 1.6 ms (corresponding to Helmholtz resonance with valve included).

Conclusion

This Brief illustrates several powerful features in ADINA FSI, including contact, gap and moving wall conditions, and the glue mesh feature. It also demonstrates ADINA's ability to help explain complex results emerging from coupled behaviors.

A tutorial outlining the steps required to perform the above FSI analysis is given below (available only to our ADINA users):